Illegal immigrants and their supporters said Thursday they will keep up a continuous vigil outside the White House until President Obama agrees to stop deportations, as they seek to ramp up pressure on the White House.

It’s the latest move from immigrant-rights advocates who have despaired of getting a legalization bill through Congress this year, and are instead calling on Mr. Obama to unilaterally halt most deportations in order to create at least a tentative legalization.

Beginning Saturday, activists will station themselves in Lafayette Park, across the street from the White House, and have vowed to stay until the president comes out to meet with them and talk to illegal immigrants and those whose relatives have already been deported.

“We have seen a lot of folks within both parties play with the lives of the undocumented, many times for political reasons,” said Erika Andiola, one of the activists, standing with others on Pennsylvania Avenue outside the White House. “We are going to stay outside of your house, we’re going to stay outside of your door, your window, until you stop the deportations of our own families.”

Ms. Andiola is an illegal immigrant who was granted tentative legal status under Mr. Obama’s nondeportation policies, and even landed a job working for a member of Congress using her new work permit. She has since left that job to advocate on behalf of her family — her mother and brother, who have not earned similar tentative status, are both in deportation proceedings.

Mr. Obama’s rate of deportations has become a hot political topic in recent months.

According to calculations from The Washington Times, the president will top 2 million deportations for his tenure on April 9. That would be a record rate — though the number isn’t exactly comparable to previous administrations because the Obama administration counts border removals in its deportation statistics.

With the actual numbers unclear, Mr. Obama has taken fire from both sides.

Activists call him “deporter-in-chief” and vow to punish his party at the polls in November.

But conservatives in Congress argue that far from setting records, Mr. Obama is refusing to enforce the law in many cases. Indeed, his former chief of Immigration and Customs Enforcement told the Los Angeles Times that the chances of a “run-of-the-mill” illegal immigrant living in the U.S. interior being deported were almost zero.

Fearing a disheartened Democratic base in the upcoming elections, Mr. Obama has promised a new review to try to find other categories of illegal immigrants he can stop deporting.

The immigration issue is heating up on Capitol Hill, too, where some congressional Republicans are searching for a small legalization bill they can pass in order to try to argue they are tackling at least part of the problem.

Some Republicans are trying to attach to the annual defense policy bill a legalization program for young adult illegal immigrants who are willing to join the military. They would eventually be eligible for citizenship.

That proposal has left deep divisions among House Republicans, while activist groups said it was a small beginning, but didn’t go far enough.

Rep. Mike Coffman, Colorado Republican and one of those pushing for the proposal, said it applies to “children who were brought into this country illegally through no fault of their own, and are willing to stand up and serve in the military.”

“As a Marine combat veteran, I believe there is no higher expression of citizenship than serving our nation in uniform,” he said.

But the kind of broad legalization for the estimated 11-12 million illegal immigrants already in the country appears dead in the GOP-led House. Unified Republicans on the House Budget Committee on Wednesday defeated an effort to add a full legalization to the 2015 budget.